Taylor's frozen-flow hypothesis in Burgers turbulence

By detailed analytical treatment of the shock dynamics in the Burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space. Our analytical treatment verifies the so-called Taylor's frozen-flow hypothesis without relying on any closure and under very general assumptions. We discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function. We support the analytical calculation by performing numerical simulation of the periodically kicked Burgers equation.

Figure 1

(Color online) The scaling of the time increments of velocity $⟨{\omega }^q⟩$ verses $t_2-t_1=t$ is sketched for different moments $q∊(0,2]$. The kicking time is ${10}^{-3}$ and the measured time span is showing the scaling both below and above it.

Figure 2

(Color online) The saturation of ${\zeta }_q$ for different $q$’s, where ${\zeta }_q$ is the scaling exponents $⟨{\mid \omega \mid }^q⟩\sim {\mid t\mid }^{{\zeta }_q}$. The upper and lower figures are the saturation of scaling exponents for the time span smaller and bigger than kicking time, respectively. The correlation time scale is about $t_{\mathit{corr}}\simeq 3\times t_{\mathit{\text{kick}}}$. The results shows that Taylor frozen hypothesis is preserved just for time scales less than $t_{\mathit{corr}}$.

Figure 3

(Color online) The conditional moment $⟨u^2\mid \omega ⟩$ as a function of $\omega$ at different time spans $\mid t_2-t_1\mid$ below and above the kicking time. For any given time increment $t$ the conditional average is normalized by the corresponding $⟨u^2⟩$.